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प्रश्न
If y = 2x is a chord of circle x2 + y2−10x = 0, find the equation of circle with this chord as diametre
उत्तर
The equation of the circle is
x2 + y2 − 10x = 0 ...(1)
and the equation of the line is y = 2x ...(2)
Let A(x1, y1) and B(x2, y2) be the points of intersection of circle and the line.
To find the points of intersection, substitute y = 2x in 1 equation (1), we get,
x2 + 4x2 − 10x = 0
∴ 5x2 − 10x = 0
∴ x(5x − 10) = 0
∴ x = 0 or x = 2
∴ x1 = 0 and x2 = 2
Also, y1 = 2x1 and y2 = 2x2
∴ y1 = 0 and y2 = 4
∴ A ≡ (0, 0) and B ≡ (2, 4)
∴ by diameter form, the equation of the circle on chord AB as diameter is
(x − 0)(x − 2) + (y − 0)(y − 4) = 0
∴ x2 − 2x + y2 − 4y = 0
∴ x2 + y2 − 2x − 4y = 0.
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